We study the dynamics of a point particle in an array of spherical scatterers centered at the points of a quasicrystal (of cut-and-project type). It turns out that, as in the case of a periodic scatterer configuration, the stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit) is Markovian after an appropriate extension of the phase space. The definition and properties of the transition kernel for this limiting Markov process are more complicated in the quasicrystal case, and we will discuss this both in general and in specific examples. Homogeneous dynamics enters as a key tool in the proofs. This is joint work with Jens Marklof.